A049130 Revert transform of ((x - 1)(x + 1))/(-1 - x + x^3).

1, 1, 2, 4, 10, 26, 73, 211, 630, 1918, 5944, 18668, 59311, 190243, 615269, 2004025, 6568174, 21645438, 71681152, 238416580, 796107464, 2667768904, 8968626418, 30240087086, 102238147891, 346514952331, 1177137322768, 4007326361986, 13669068510355, 46711170248183, 159899495303170

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

Binomial transform of aerated version of A001764. - Paul Barry, Nov 05 2006

a(n) is the number of planar rooted trees with n vertices, where each vertex has one or an even number of children. - Kurusch Ebrahimi-Fard and Dominique Manchon (kef(AT)unizar.es), Jun 05 2010

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1808
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• K. Ebrahimi-Fard and D. Manchon, Dendriform Equations, Journal of Algebra, 322 (2009), 4053-4079.
• Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
• Index entries for reversions of series

Programs

Maple

series(RootOf((x-1)*A^3+(1-x)*A-x,A),x=0,30); # Mark van Hoeij, Apr 16 2013

Mathematica

Table[Sum[Binomial[n,2*k]*Binomial[3*k,k]/(2*k+1),{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Jun 28 2013 *)

PARI

Vec( serreverse(x*(1-x)*(1+x)/(1+x-x^3) +O(x^66) ) ) \\ Joerg Arndt, Sep 12 2013

Formula

a(n) = sum{k=0..floor(n/2), C(n,2k)*C(3k,k)/(2k+1)}; a(n)=sum{k=0..n, C(3k/2,k/2)(1+(-1)^k)/(2(k+1))}. - Paul Barry, Nov 05 2006

G.f.: A(x) satisfies (x-1)*A(x)^3+(1-x)*A(x)-x = 0. - Mark van Hoeij, Apr 16 2013

Representation in terms of special values of hypergeometric function of type 4F3, in Maple notation a(n) = hypergeom([-n/2, (1-n)/2, 1/3, 2/3], [1, 1/2, 3/2], 27/4), n=0,1,2... . - Karol A. Penson, Jun 20 2013

Recurrence (for offset 0): -23*(n-3)*(n-2)*a(n-4) + 19*(n-2)*(2*n-3)*a(n-3) - 3*(n-2)*n*a(n-2) - 8*n*(2*n-1)*a(n-1) + 4*n*(n+1)*a(n) = 0. - Vaclav Kotesovec, Jun 28 2013

Asymptotics (for offset 0): a(n) ~ sqrt(3/Pi)*sqrt(1/2+1/(3*sqrt(3)))^3 * (1+3/2*sqrt(3))^n/n^(3/2). - Vaclav Kotesovec, Jun 28 2013

G.f.: 1/(1-x - 3*x^2/(S(0)*(1-x))), where S(k) = 4*k+3 - 3*x^2*(3*k+4)*(6*k+5)/( 2*(1-x)^2*(4*k+5) - 3*x^2*(3*k+5)*(6*k+7)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2013